Problem 63 - Chemistry and Chemical Reactivity

Question

You have four gas samples: 1\. \(1.0 \mathrm{L}\) of \(\mathrm{H}_{2}\) at \(\mathrm{STP}\) 2\. \(1.0 \mathrm{L}\) of \(\mathrm{Ar}\) at \(\mathrm{STP}\) 3\. \(1.0 \mathrm{L}\) of \(\mathrm{H}_{2}\) at \(27^{\circ} \mathrm{C}\) and \(760 \mathrm{mm}\) Hg 4\. \(1.0 \mathrm{L}\) of He at \(0^{\circ} \mathrm{C}\) and \(900 \mathrm{mm} \mathrm{Hg}\) (a) Which sample has the largest number of gas particles (atoms or molecules)? (b) Which sample contains the smallest number of particles? (c) Which sample represents the largest mass?

Step-by-Step Solution

Answer

Sample 4 has the most particles. Sample 3 has the fewest particles. Sample 2 has the largest mass.

Step 1: Understanding Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) refers to a condition of 0°C (273.15 K) temperature and 1 atmosphere (760 mm Hg) pressure. At STP, 1 mole of any gas occupies a volume of 22.4 L.

Step 2: Identify the Conditions for Each Gas Sample

- Sample 1: H₂, 1.0 L at STP - Sample 2: Ar, 1.0 L at STP - Sample 3: H₂, 1.0 L at 27°C (300.15 K) and 760 mm Hg - Sample 4: He, 1.0 L at 0°C (273.15 K) and 900 mm Hg.

Step 3: Using Avogadro's Law for Number of Particles

According to Avogadro's Law, equal volumes of gases at the same temperature and pressure contain equal numbers of particles. Thus, Samples 1 and 2 have the same number of particles as they are both at STP.

Step 4: Calculate Moles for Sample 3 Using Ideal Gas Law

The ideal gas law is given by \[ PV = nRT \]where \( P = 760 \text{ mm Hg} = 1 \text{ atm} \), \( V = 1.0 \text{ L} \), \( R = 0.0821 \text{ L atm K}^{-1}\text{ mol}^{-1} \), \( T = 300.15 \text{ K} \).Solve for \( n \):\[ n = \frac{PV}{RT} = \frac{(1)(1)}{0.0821 \times 300.15} ≈ 0.0406 \text{ mol} \]

Step 5: Calculate Moles for Sample 4 Using Ideal Gas Law

For Sample 4, using \( P = \frac{900}{760} \text{ atm} \), \( V = 1.0 \text{ L} \), \( R = 0.0821 \text{ L atm K}^{-1}\text{ mol}^{-1} \), \( T = 273.15 \text{ K} \):\[ n = \frac{PV}{RT} = \frac{(\frac{900}{760})(1)}{0.0821 \times 273.15} ≈ 0.0483 \text{ mol} \]

Step 6: Determine Largest and Smallest Number of Particles

From calculations, Sample 4 has the largest number of moles (0.0483 mol) and thus the largest number of particles. Sample 3 has the smallest number of moles (0.0406 mol) and thus the smallest number of particles.

Step 7: Calculate Mass for Each Sample

- Sample 1: \(1.0 \text{ mol H}_2\). Molar mass = 2 g/mol. Mass = 0.0446 mol * 2 g/mol ≈ 0.0892 g.- Sample 2: \(0.0446 \text{ mol Ar}\). Molar mass = 39.95 g/mol. Mass = 0.0446 mol * 39.95 g/mol ≈ 1.78 g.- Sample 3: As calculated above, it contains 0.0406 mol of H₂ which is lighter than Ar.- Sample 4: \(0.0483 \text{ mol He}\). Molar mass = 4 g/mol. Mass = 0.0483 mol * 4 g/mol ≈ 0.1932 g.

Step 8: Determine the Sample with the Largest Mass

From the calculations, Sample 2 (argon) has the largest mass of approximately 1.78 g.

Key Concepts

Avogadro's Law

Avogadro's Law is a fundamental concept in chemistry that helps us understand the relationship between the volumes of gases and the number of particles they contain. It states that at a constant temperature and pressure, equal volumes of all gases contain the same number of molecules. This means that if you have two gases, like hydrogen and argon, occupying the same volume under identical conditions of temperature and pressure, they will have the same number of particles.
For the textbook exercise, samples 1 and 2 both occupy 1 liter at Standard Temperature and Pressure (STP). Applying Avogadro's Law, they each contain the same number of gas particles, despite being different types of gases. This highlights the simplicity and uniformity Avogadro's Law brings to comparing gas samples under equivalent conditions. When using Avogadro's Law, remember that it assumes ideal gas behavior, which is a good approximation for most gases at normal temperatures and pressures.

Standard Temperature and Pressure (STP)

Understanding Standard Temperature and Pressure (STP) is crucial in studying gases and conducting experiments. STP is defined as a temperature of 0°C (273.15 K) and a pressure of 1 atmosphere (760 mm Hg). At these conditions, one mole of an ideal gas occupies a volume of 22.4 liters. These 'standard' conditions provide a reference point to compare gas behavior.
In the exercise, samples 1 and 2 are both measured at STP. This allows us to directly apply Avogadro's Law since both samples occupy the same volume of 1 liter at STP. Therefore, they contain the same number of particles. Standardizing temperature and pressure helps scientists communicate findings and compare data in a consistent and reliable manner. Although actual experimental conditions often vary from STP, knowing these standards is essential for calculations and understanding gas laws.

Calculating Moles

Calculating moles is a vital part of solving gas problems, especially when dealing with ideal gases. Moles provide a way to quantify the amount of substance, connecting it to particle number through Avogadro's number. The Ideal Gas Law, expressed as \( PV = nRT \), allows us to rearrange and solve for \( n \) (the number of moles), provided we know the pressure \( P \), volume \( V \), and temperature \( T \) of the gas, along with the gas constant \( R \).
In the original exercise, Sample 3 and Sample 4 were evaluated using the Ideal Gas Law due to their non-STP conditions. For Sample 3, at 27°C and 760 mm Hg, we found it had approximately 0.0406 moles of hydrogen. For Sample 4, at 0°C and 900 mm Hg, it held about 0.0483 moles of helium. Solving for moles gives us insight into how many gas particles are present in each sample. This calculation is essential to identifying which samples have the largest or smallest number of particles.

Gas Particle Number

The number of particles in a gas, which can be atoms or molecules, determines many of its properties and behaviors. Knowing the number of gas particles in a sample is important for understanding reactions, computing pressures, and calculating volumes. Using the mole concept along with Avogadro's number, we can convert between moles and particles, where one mole equals approximately \(6.022 \times 10^{23}\) particles.
In the given exercise, after determining the number of moles for each gas sample, we were able to establish the number of particles they contained. Sample 4, which contained helium, had the most particles due to its higher mole count (0.0483 moles), while Sample 3, with hydrogen at 0.0406 moles, had the least. Understanding the number of gas particles allows chemists to predict how gases will behave under various conditions and facilitates calculations involving chemical reactions and gas-phase processes.